Question

If \(f(x)=a^{x},\) show that \(f(\alpha x)=[f(x)]^{\alpha}\).

Short answer

\( f(\alpha x) = [f(x)]^{\alpha} \)

Step-by-step solution

- Recall the Definition of the Function

Begin by recalling the definition of the function given in the problem, which is \( f(x) = a^x \).

- Substitute \(\alpha x\) into the Function

Substitute \( \alpha x \) for \( x \) in the function \( f(x) = a^x \). This gives us: \( f(\alpha x) = a^{\alpha x} \).

- Use Properties of Exponents

Recall the property of exponents that states \( a^{m \times n} = (a^m)^n \). Apply this property to \( a^{\alpha x} \): \( f(\alpha x) = a^{\alpha x} = (a^x)^{\alpha} \).

- Relate Back to \(f(x)\)

Recognize that \( a^x \) is the same as \( f(x) \). Substitute \( f(x) \) back into the expression: \( f(\alpha x) = (a^x)^{\alpha} = [f(x)]^{\alpha} \). Hence, we have shown that \( f(\alpha x) = [f(x)]^{\alpha} \).

Key concepts

Function Definition

In mathematics, a function is an expression that relates inputs to outputs. Specifically, our function is defined as \( f(x) = a^x \). This type of function is called an exponential function because it involves an exponent with a variable base. The base \(a\) is a constant, and \(x\) is the variable. These functions are powerful tools in modeling exponential growth or decay, such as population growth or radioactive decay.

To understand the function definition, remember that the base must be a positive real number other than 1. For example, if our base is 2, then the function becomes \( f(x) = 2^x \). Simple enough!

Substitution

Substitution is a helpful technique in algebra for solving and simplifying equations. In our problem, to show that \(f(\alpha x) = [f(x)]^{\alpha}\), we begin by substituting \( \alpha x \) for \( x \) in our function definition.

Given the function \( f(x) = a^x \), substitute \( \alpha x \) to find \( f(\alpha x) \). This substitution gives us:

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